While using style="font-family:..." (or the deprecated fontfamily="..." attribute, which does work in SVG) will cause MathJax to try to us the given font for whatever characters it can, this will not produce very good results, for several reasons: first, it relies on the user having the given font installed locally on their own computer (MathJax does not try to use web fonts for these, though you could set up CSS to do that yourself). Second, MathJax does not have the font metrics for the given font, and so the position of super and subscripts, and other layout will not be as accurate as usual. Third, MathJax will try to measure the widths of the characters, but the mechanism for this will cause a reflow of the page for each character, and that will have a significant negative impact on the performance of MathJax's typesetting.
For these reasons, I don't recommend using font-family CSS except for isolated characters for special needs.
As for running MathJax once with one font configuration and then a second time with another, that is a good idea, but unfortunately, MathJax only loads one set of fonts, and changing the configuration will not cause it to load another. So you won't be able to use that approach. The only way I can see that you could do it would be to change the output jax as well (since each output jax can have a different font), or to delete the MathJax global variable and reload MathJax again from scratch with the new configuration. Both of these seem to be rather heavy-handed approaches.
Davide